According to rules , wee are allow to do only one question at a time so i do only question 8.. you post others separately.. so that i will able to help you.
Exercise 6: Given the table of the function f(x)-2" 2 X 0 3 2 f(x) 1 2 4 8 a) Write down the Newton polynomial...
7. (15 pts) Numerical Integration. Given a continuous function f (x) on the interval [a, b], write the Lagrange form of the quadratic polynomial interpolating f(a), (a b)), f(b). Instead of calculating the integral I(f) Jaf(x)dx we could approximate it via Q(f) = | q(x)dx. Find an expression for this quadrature rule, the so-called Simpson's rule.
Show work by hand and also using MATLAB code. Model 1 Given a polynomial f(x) Write a first-order approximation of f(x), given the value of f(x) at two points Plot the polynomial and the first-order approximation on a graph Write a second-order approximation of f(x), given the value at three points. Plot the polynomial, the first-order and second-order approximations on a graph Find the integral Exactly Using trapezoidal rule Using composite trapezoidal rule Using Simpson's 1/3 rule . Calculate the...
The nth-order Taylor polynomial for a function f(x) using the h notation is given as: Pa (x + h) = f(x) + f'(a)h + salt) 12 + () +...+ m (s) n." The remainder of the above nth-order Taylor polynomial is defined as: R( +h) = f(n+1)(C) +1 " hn+1, where c is in between x and c+h (n+1)! A student is using 4 terms in the Taylor series of f(x) = 1/x to approximate f(0.7) around x = 1....
please answer question 5. question 4 is provided for reference. Problem 5: Application of composite integration rules Assume that you want to approximate the integral of a function in [0, 21 and have only the values of f on specific x values, given in the following table. x005 11.5| 2 (a) Find an approximation to f(x) dx using the composite trapezoid integration rule. Specify all parameters of the approximation and carry out the calculation completely. Assignment 8 MATH363, Spring 2019...
You are given the table below. 16 20 4 8 12 X f(x) 12 2417 6 30 Use the table and n = 4 to estimate the following. Because the data is not monotone (only increasing or only decreasing), you should sketch a possible graph and draw the rectangles to ensure you are using the appropriate values for a lower estimate and an upper estimate. 20 f(x)dx lower estimate upper estimate Estimate the area of the region under the curve...
π/2 (6 3 cos x) dx 0 (a) Derive the formula for multi-segment (evenly spaced) left-hand rectangles and then use it to approximate the value of the integral with n=1; n-2; n-4 segments. Calculate the true error and relative true error for each (b) Derive the formula for multi-segment (evenly spaced) right-hand rectangles and then use it to approximate the value of the integral with n=1; n=2; n-4 segments. Calculate the true error and relative true error for each (c)...
QUESTION 5 The integral 2 1 I= dx x +4 0 is to be approximated numerically. (a) Find the least integer M and the appropriate step size h so that the global error for the composite Trapezoidal rule, given by -haf" (), a< & <b, 12 is less than 10-5 for the approximation of I. b - a (b) Use the two-term Gaussian quadrature formula and 6 decimal place arithmetic to approximate I. (Hint: Parameters are ci = 1, i...
6. (20) Consider the table X 02 3 y = f(x) 7 11 -4 (a) Fit the given data with a quadratic spline. (b) Find an approximate value of f(3) using the first-degree spline. 6. (20) Consider the table X 0 | 2 | y = f(x) | 7 11 | 3 5 (a) Fit the given data with a quadratic spline. YE (b) Find an approximate value of f (3) using the first-degree spline.
Problem 3. Suppose you are programming the composite trapezoid rule (CTR) to approximate 1(f) =| f(x) dx using the TR with N subintervals, and that you mistakenly forget to weight down the two endpoints by 3. That is, you have accidentally programmed the quadrature rule where h-%.. (Note: sinoefe C, you know that UIL is bounded.) 1. Find QBADN -OCTRN where QCTRN ) is the approximation to (x) dx computed via the CTR with N subintervals. Problem 3. Suppose you...
4. Given a function f(x), use Taylor approximations to derive a second order one-sided ap- proximation to f'(ro) is given by f(zo + h) + cf (zo + 21) + 0(h2). f' (zo) = af(xo) + What is the precise form of the error term? Using the formula approximate f' (1) where r) = e* for h 1/(2p) for p = 1 : 15, Form a table with columns giving h, the approximation, absolute error and absolute error divided by...